
reduced if \(N(R)=0\) .

reversible ([1]}) if \(wh=0\)  implies \(hw=0\)  for any \(w,~h\in R\) .

semicommutative ([1]}) if for each \(w\in R\) , \(r(w)\)  is an ideal of \(R\) .

strongly regular ([3]}) if for each \(w\in R\) , \(w\in w^2R\) .

left (right) weakly regular ([4]}) if \(w\in RwRw~ (w\in wRwR)\)  for any \(w\in R\) .

left (right) quasi duo ([3]}) if every maximal left (right) ideal of \(R\)  is an ideal of \(R\) .
   Let \(ME_l(R)=\lbrace e\in E(R)~|~Re\)   is  a minimal left ideal of \(R\rbrace \) . \(R\)  is called left min-abel if for any \(e\in ME_l(R)\)  \(re=ere\)  for all \(r\in R\) . \(R\)  is called left \(MC2\)  if \(aRe=0\)  implies \(eRa=0\)  for any \(a\in R,~e\in ME_l(R)\) . According to [6]}, \(R\)  is said to be \(NCI\)  if \(N(R)=0\)  or \(N(R)\)  contains a non-zero ideal of \(R\) . \(R\)  is said to be \(NI\)  if \(N(R)\)  is an ideal of \(R\) , and \(R\)  is 2-primal if \(N(R)=P(R)\) . Obviously, NI rings are NCI; nevertheless, the converse is not true (by [6]}). \(R\)  is directly finite if \(wh=1\)  implies \(hw=1\)  for any \(h,w\in R\) .
   Let \(\Psi :R\rightarrow R\)  be an automorphism of \(R\) . \(R[x;\Psi ]\)  is the ring of polynomials over \(R\)  with respect to usual polynomial addition and multiplication which is defined by the rule: \(xa=\Psi (a)x\) . \(R[x;\Psi ]\)  is called skew polynomial ring of \(R\) .
   Over the past several years, semicommutative rings and their generalizations have been studied extensively by many authors. In a semicommutative ring \(R\) , both the left and the right annihilator of every element of \(R\)  are ideals of \(R\) . This motivates us to investigate a ring \(R\)  in which either the left or the right annihilator (not necessarily both) of any element of \(R\)  is an ideal of \(R\) . This paper studies such a class of rings.

